L11n71

L11n70

L11n72

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Link Presentations

[edit Notes on L11n71's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,16,8,17 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X15,8,16,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uv^{3}-uv^{2}-2uv+2u+2v^{3}-2v^{2}-v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{23/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+2za^{11}+a^{11}z^{-1}+3za^{9}+2a^{9}z^{-1}-z^{5}a^{7}-5z^{3}a^{7}-5za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-4z^{3}a^{5}-2za^{5}$ (db)
Kauffman polynomial	$a^{14}z^{8}-7a^{14}z^{6}+16a^{14}z^{4}-14a^{14}z^{2}+4a^{14}+a^{13}z^{9}-6a^{13}z^{7}+10a^{13}z^{5}-5a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+3a^{12}z^{8}-20a^{12}z^{6}+41a^{12}z^{4}-33a^{12}z^{2}+9a^{12}+a^{11}z^{9}-5a^{11}z^{7}+3a^{11}z^{5}+5a^{11}z^{3}-2a^{11}z-a^{11}z^{-1}+2a^{10}z^{8}-13a^{10}z^{6}+22a^{10}z^{4}-13a^{10}z^{2}+4a^{10}+2a^{9}z^{7}-14a^{9}z^{5}+24a^{9}z^{3}-13a^{9}z+2a^{9}z^{-1}+a^{8}z^{6}-7a^{8}z^{4}+8a^{8}z^{2}-2a^{8}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-8a^{7}z+2a^{7}z^{-1}+a^{6}z^{6}-4a^{6}z^{4}+2a^{6}z^{2}+a^{5}z^{5}-4a^{5}z^{3}+2a^{5}z$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     1 1 0 -8                 1 1     0 -10               1 1 1     1 -12             2 4 1       1 -14           1   1         2 -16         1 3 2           0 -18       2 1               1 -20       1 1               0 -22   1 2                   -1 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-7$ ${\mathbb {Z} }$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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